A Tale of Three Intersecting Lines: questions with answers

A triangle’s interior angles total 180 degrees. For either base length, the third angle is $180^\circ - 35^\circ - 50^\circ = 95^\circ$. Since both base angles stay the same, the third angle is unchanged.

The three interior angles of a triangle add to 180 degrees, and each is greater than 0 degrees. The second and third angles together equal 180 − 54 = 126 degrees. So the second angle can be any value greater than 0 degrees and less than 126 degrees. At 126 degrees, the third angle would be 0 degrees, so a triangle would not exist.

An interior angle of a triangle must be greater than 0 degrees and less than 180 degrees. With two positive sides, any angle in that range gives a triangle. At 180 degrees the sides lie on one straight line. Larger angles cannot be interior angles of a triangle. Apply this test separately to 30 degrees, 200 degrees and 360 degrees.

The foot lies outside the base segment exactly when one base angle is obtuse. If both base angles are acute, the foot lies between the endpoints. If a base angle is right, the foot is that endpoint. An obtuse angle at the opposite vertex does not put this altitude outside the base.

The base BC is horizontal, so the perpendicular from A to its line is vertical. In this drawing, that vertical line meets the base line outside segment BC: Beyond C, on the base extended past C. An altitude may meet the extension of a side rather than the side segment itself.

The adjacent interior and exterior angles share an arm. Their other arms form a straight line, so the two angles together make 180 degrees. The exterior angle is therefore 180 - 35 = 145 degrees. They would be equal only for a 90-degree interior angle.

The exterior angle equals the sum of the two remote interior angles. Both that sum and the exterior angle equal 180 degrees minus the adjacent interior angle. Equality with the adjacent interior angle occurs only when both are 90 degrees, so it is not a rule for every triangle.

The three adjacent angles exactly fill a straight angle, so they total 180 degrees. The top interior angle is 180 - 20 - 60 = 100 degrees. The parallel line gives the usual triangle-angle-sum construction; no measurement or line length is needed.

The proposed angles total 135 degrees. A triangle’s interior angles total exactly 180 degrees. Three equal interior angles therefore each measure 60 degrees, regardless of the triangle’s size. The required change is: Increase by 45 degrees.

ABC and BCA name the same three vertices. A triangle can be named starting at any vertex and going in either direction. So BCA is another name for triangle ABC.

Use radii 6 cm and 8 cm. Each arc fixes the distance from one base endpoint to the third vertex, so its radius is one of the two remaining side lengths. The radii need not be equal.

The radii equal the lengths of the two sides meeting at the third vertex. They are equal exactly when those two sides are equal. This includes an equilateral triangle and an isosceles triangle with its equal sides meeting at that vertex. Equal radii do not make circles with distinct centres coincide.

Both triangles share the base and the two circle radii as their side lengths. They are mirror images across the base, so their shapes, sizes and areas match. Two placements do not mean two non-congruent triangles.

AC and AD are radii of the circle with centre A, so they are equal. BC and BD are radii of the circle with centre B, so they are equal. AB is the same base for both triangles. D lies below AB, so D is the required third vertex.

Exactly one comparison is false. Each of the two shorter lengths is less than the longest length plus another positive length. Since no triangle is possible, the longest length is equal to or greater than the sum of the other two.